Plz no cursive 4 use spherical Cirardinates to find thevolumi of he rand outside z=ty" andxY-planeolid inside xty442²-9above the

Name: Plz no cursive 4 use spherical Cirardinates to find thevolumi of he ra
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Description: Plz no cursive 4 use spherical Cirardinates to find thevolumi of he rand outside z=ty" andxY-planeolid inside xty442²-9above the

Given sphere, x2+y2+z2=9 outside z=x2+y2 and above xy plane. We use spherical coordinates to find…

Answered: H use spherical Cingidinates to find…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Plz no cursive

4 use spherical Cirardinates to find the
volumi of he r
and outside z=ty" and
xY-plane
olid inside xty442²-9
above the

Expert Solution

Step 1

Given sphere, $x^{2} + y^{2} + z^{2} = 9 o u t s i d e z = \sqrt{x^{2} + y^{2}} a n d a b o v e x y p l a n e .$

We use spherical coordinates to find volume.

we know,

$\begin{array}{rcl} x^{2} + y^{2} + z^{2} & = & p^{2} \\ \Rightarrow & p^{2} = 9 \\ \Rightarrow & p = 3 \\ x & = & p \sin \emptyset \cos θ \\ y & = & p \sin \emptyset \sin θ \\ z & = & y \cos \emptyset \\ W e h a v e z & = & \sqrt{x^{2} + y^{2}} \\ p \cos \emptyset & = & \sqrt{p^{2} \sin^{2} \emptyset \cos^{2} θ + p^{2} \sin^{2} \emptyset \sin^{2} θ} \\ = & \sqrt{p^{2} \sin^{2} \emptyset (\cos^{2} θ + \sin^{2} θ)} \\ = & \sqrt{p^{2} \sin^{2} \emptyset (1)} \\ p \cos \emptyset & = & p \sin \emptyset \\ \cos \emptyset & = & \sin \emptyset (∵ \sin θ = \cos θ, w h e n θ = \frac{π}{4}) \\ \emptyset & = & \frac{π}{4} \end{array}$

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